Stable solution and extremal solution for fractional $ p $-Laplacian

To our knowledge, this paper is the first attempt to consider the existence issue for fractional $ p $-Laplacian equation: $ (-\Delta)_p^s u = \lambda f(u), \; u> 0 \; \text{in}\; \Omega;\; u = 0\;\text{in}\; \mathbb{R}^N\setminus\Omega $, where $ p>1 $, $ s\in (0, 1) $, $ \lambda>0 $ and $ \Omega $ is a bounded domain with $ C^{1, 1} $ boundary. We first propose a notion of stable solution, then we prove that when $ f $ is of class $ C^1 $, nondecreasing and satisfying $ f(0)>0 $ and $ \underset{t\to \infty}{\lim}\frac{f(t)}{t^{p-1}} = \infty $, there exists an extremal parameter $ \lambda^*\in (0, \infty) $ such that a bounded minimal solution $ u_\lambda \in W_0^{s, p}(\Omega) $ exists if $ \lambda\in (0, \lambda^*) $, and no bounded solution exists if $ \lambda>\lambda^* $. Moreover, no $ W_0^{s, p}(\Omega) $ solution exists for $ \lambda > \lambda^* $ if in addition $ f(t)^{\frac{1}{p-1}} $ is convex.

To handle our problems, we show a Kato-type inequality for fractional $ p $-Laplacian. We show also $ L^r $ estimates for the equation $ (-\Delta)_p^su = g $ with $ g\in W_0^{s, p}(\Omega)^*\cap L^q(\Omega) $ for $ q \geq 1 $, especially for $ q \le \frac{N}{sp} $. We believe that these general results have their own interests. Finally, using the stability of minimal solutions $ u_\lambda $, under the polynomial growth or convexity assumption on $ f $, we show that the extremal function $ u_* = \lim_{\lambda\to\lambda^*}u_\lambda \in W_0^{s, p}(\Omega) $ in all dimensions, and $ u^*\in L^{\infty}(\Omega) $ in some low dimensional cases.